Worry about it - no. Wonder about it - yes. Cruise lines have always asked this. Have you had a fever, have you had stomach illness, etc. My thought is who is going to say "yes". The cruise lines tell you this well in advance, so if you are the type of person who would not cruise because they had recently been ill, then they are not going to show up at port. If they are at the port, then they are planning on getting on board, and are likely to lie to do so. How many people say "OMG, you mean if I was recently sick, I should not get on a ship full of other people in close proximity! I had no idea. Of course let me cancel my cruise."
We have a cruise booked for the end of July (on Celebrity, not DCL) and right now we are taking a wait and see attitude. Celebrity is allowing anybody with a cruise departing up to July 31 to cancel with 48 hours notice. You can apply your money to a new cruise or get a refund. We are leaving from Bayonne NJ, so we just drive over to the cruise port, so we have no worries about airfare, hotels and other such things.
We are booked in Hilton Head DVC for the end of June, that we are leery about. (For the record, we are not alarmists, my wife just happens to be immune compromised and has asthma - so we got that going for us)